🔗 Conditional Probability

Definition

Conditional probability: Probability of $A$ given $B$ has occurred

$P(A|B) = \frac{P(A \cap B)}{P(B)}$

Intuition: Restrict sample space to outcomes where $B$ occurred; within that, find proportion where $A$ also occurred

Notation: "$P(A|B)$" reads as "probability of $A$ given $B$"

Worked Example with Two-Way Table

Data: 200 AP Statistics students

Question: What's the probability a student passed, given they attended class?

$P(\text{Passed} | \text{Attended}) = \frac{P(\text{Passed} \cap \text{Attended})}{P(\text{Attended})} = \frac{95/200}{100/200} = \frac{95}{100} = 0.95$

Question: What's the probability a student attended class, given they passed?

$P(\text{Attended} | \text{Passed}) = \frac{95/200}{135/200} = \frac{95}{135} ≈ 0.704$

Note: $P(A|B) \neq P(B|A)$

Tree Diagrams

For sequential events:

Start
├─ A (prob=0.4)
│  ├─ C (prob=0.7)
│  └─ D (prob=0.3)
└─ B (prob=0.6)
   ├─ C (prob=0.2)
   └─ D (prob=0.8)

Path probabilities: Multiply along branches

• $P(A \text{ and } C) = 0.4 × 0.7 = 0.28$
• $P(B \text{ and } C) = 0.6 × 0.2 = 0.12$

Total: $P(C) = 0.28 + 0.12 = 0.40$

Multiplication Rule (General)

$P(A \cap B) = P(A) · P(B|A)$

Or equivalently: $P(A \cap B) = P(B) · P(A|B)$

Use: To find probability of both events

Common Mistakes

• Confusing $P(A|B)$ with $P(B|A)$
• Using $P(A \cap B)$ when conditional probability asked
• Forgetting to divide by $P(B)$ in conditional probability formula

Decision Rule

"Given that" in problem? → Use conditional probability Computing $P(A \text{ and } B)$ first? → Use multiplication rule

AP Exam Tip

Always redraw two-way table or tree for conditional probability problems. Shading/circling the condition helps avoid errors.